Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

SOLƲTION.

If NO be made unity, NP=q, and NQ be supposed =z; having drawn QS parallel to TO, you'l have three similar triangles NOQ, QNR and RQS. For since the an∣gle QOP is double of the angle QON, and the same (as be∣ing at the Center) double also of the angle at the Periphery QNR; it will be equal to the angle QON. But the angle at Q is common to both triangles: Therefore the whole are equi-angular, and consequently the legs NQ and NR equal, as also NO and QO; and by the like reason also PY and PT. Wherefore if RS should be added to RY, the line NP by this addition would be triple of the line NQ; and so would give the Equation, if RS was determined; which may be done by means of the ▵ QRS, similar to the two former NOQ and QNR; for the angle RQS is equal to the alternate one QOP=QNR, and the angle at R common to the tri∣angles QNR and RQS, &c. Wherefore as NO to NQ so NQ to QR 1−z−z−zz and as NQ to QR so QR to RS z−zz−zz−〈 math 〉〈 math 〉

Therefore according to what we have above said 〈 math 〉〈 math 〉; and substracting 〈 math 〉〈 math 〉; or 〈 math 〉〈 math 〉

Therefore the Central Rule will be (supposing also unity NO for the Latus Rectum,)

〈 math 〉〈 math 〉 = AD i. e. by our ½+〈 math 〉〈 math 〉 i. e. 2NO=AD form, and 〈 math 〉〈 math 〉=DH. and 〈 math 〉〈 math 〉=DH.

Geometrical Construction. Having described your parabola (Fig. 51. n. 2.) take on its Ax (because the quantity p is want∣ing in the Equation) AD=2NO, and from D having e∣rected a perpendicular = 〈 math 〉〈 math 〉NP to H; that will be the Center, from which a circle described thro' A, by cutting the parabola in three places, will give the three roots of the Equation, viz. NO and no true ones, the first whereof will express the quantity sought NQ (n. 1.) the latter the line NV, being the subtense of the third part of the compl. of the arch; and MO will express the false root, which is equal to the former taken together: All the same as in Cartes p. 91. but here some∣what plainer and easier.